Stochastic integration with respect to fractional processes in Banach spaces

TL;DR

This paper develops a framework for stochastic integration of fractional processes in various Banach spaces, including non-UMD and non-Gaussian cases, with applications to stochastic PDEs and regularity analysis.

Contribution

It provides a novel characterization of Wiener integrals for fractional processes in broad Banach spaces, extending existing theories to non-Gaussian and non-UMD settings.

Findings

Characterization of Wiener integral domains in Banach spaces.

Conditions for measurability and continuity of stochastic convolutions.

Application to regularity of solutions to stochastic PDEs.

Abstract

In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. The class of Banach spaces that is considered includes a large variety of the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. In the article, a characterization of the domains of the Wiener integrals on both bounded and unbounded intervals is given for both scalar and cylindrical fractional processes. In general, the integrand takes values in the space of $\gamma$-radonifying operators from a certain homogeneous Sobolev-Slobodeckii space into the considered Banach space. Moreover, an equivalent characterization in terms of a pointwise kernel of the integrand is also given if the considered Banach space is isomorphic with a subspace of a cartesian product of mixed Lebesgue spaces. The results are subsequently applied to stochastic convolution for which both necessary and sufficient conditions for measurability and sufficient conditions for continuity are found. As an application, space-time continuity of the solution to a parabolic equation of order $2m$ with distributed noise of low time regularity is shown as well as measurability of the solution to the heat equation with Neumann boundary noise of higher regularity.